module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.NumberTheory.Padics.PadicVal.Basic | {
"line": 156,
"column": 2
} | {
"line": 156,
"column": 50
} | {
"line": 158,
"column": 0
} | [
{
"pp": "p : ℕ\nz : ℤ\nhp : p ≠ 1\nhz : z ≠ 0\n⊢ padicValRat p ↑z = ↑(multiplicity (↑p) z)",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"congrArg",
"padicValInt",
"Rat",
"Rat.instIntCast",
"id",
"Int",
"padicVal... | [] | rw [of_int, padicValInt.of_ne_one_ne_zero hp hz] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Padics.PadicVal.Basic | {
"line": 156,
"column": 2
} | {
"line": 156,
"column": 50
} | {
"line": 158,
"column": 0
} | [
{
"pp": "p : ℕ\nz : ℤ\nhp : p ≠ 1\nhz : z ≠ 0\n⊢ padicValRat p ↑z = ↑(multiplicity (↑p) z)",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"congrArg",
"padicValInt",
"Rat",
"Rat.instIntCast",
"id",
"Int",
"padicVal... | [] | rw [of_int, padicValInt.of_ne_one_ne_zero hp hz] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Padics.PadicVal.Basic | {
"line": 156,
"column": 2
} | {
"line": 156,
"column": 50
} | {
"line": 158,
"column": 0
} | [
{
"pp": "p : ℕ\nz : ℤ\nhp : p ≠ 1\nhz : z ≠ 0\n⊢ padicValRat p ↑z = ↑(multiplicity (↑p) z)",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"congrArg",
"padicValInt",
"Rat",
"Rat.instIntCast",
"id",
"Int",
"padicVal... | [] | rw [of_int, padicValInt.of_ne_one_ne_zero hp hz] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Abelian.GrothendieckCategory.EnoughInjectives | {
"line": 283,
"column": 2
} | {
"line": 284,
"column": 50
} | {
"line": 285,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝⁷ : Category.{v, u} C\nG : C\ninst✝⁶ : Abelian C\nhG : IsSeparator G\nX : C\ninst✝⁵ : IsGrothendieckAbelian.{w, v, u} C\nA : C\nf : A ⟶ X\ninst✝⁴ : Mono f\nJ : Type w\ninst✝³ : LinearOrder J\ninst✝² : OrderBot J\ninst✝¹ : SuccOrder J\ninst✝ : WellFoundedLT J\nj : J\nhj : transfiniteIte... | [
"C : Type u\ninst✝⁷ : Category.{v, u} C\nG : C\ninst✝⁶ : Abelian C\nhG : IsSeparator G\nX : C\ninst✝⁵ : IsGrothendieckAbelian.{w, v, u} C\nA : C\nf : A ⟶ X\ninst✝⁴ : Mono f\nJ : Type w\ninst✝³ : LinearOrder J\ninst✝² : OrderBot J\ninst✝¹ : SuccOrder J\ninst✝ : WellFoundedLT J\nj : J\nhj : transfiniteIterate (larger... | refine Arrow.isoMk ((Subobject.isoOfEq _ _ (transfiniteIterate_bot _ _) ≪≫
Subobject.underlyingIso f)) (asIso t.arrow) ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.Padics.PadicVal.Basic | {
"line": 366,
"column": 2
} | {
"line": 366,
"column": 30
} | {
"line": 367,
"column": 4
} | [
{
"pp": "case cons\np j : ℕ\nhp : Fact (Nat.Prime p)\nF : ℕ → ℚ\nS : Finset ℕ\nhn1 : ∀ (i : ℕ), 0 < F i\ns : ℕ\nS' : Finset ℕ\nHnot : s ∉ S'\nHne : S'.Nonempty\nHind : (∀ i ∈ S', padicValRat p (F j) < padicValRat p (F i)) → padicValRat p (F j) < padicValRat p (∑ i ∈ S', F i)\nhF : ∀ i ∈ Finset.cons s S' Hnot, p... | [] | | cons s S' Hnot Hne Hind => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.AlgebraicGeometry.Sites.ElladicCohomology | {
"line": 55,
"column": 2
} | {
"line": 56,
"column": 85
} | {
"line": 58,
"column": 0
} | [
{
"pp": "X : Scheme\n⊢ IsGrothendieckAbelian.{u + 1, u + 1, u + 2} (Sheaf (ProEt.topology X) Ab)",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"CategoryTheory.Functor.op",
"AddCommGrpCat.FilteredColimits.forget_preservesFilteredColimits",
"CategoryTheory.GrothendieckTopo... | [] | have : EssentiallySmall.{u + 1} X.ProEt := inferInstance
exact Sheaf.isGrothendieckAbelian_of_essentiallySmall (ProEt.topology X) Ab.{u + 1} | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Sites.ElladicCohomology | {
"line": 55,
"column": 2
} | {
"line": 56,
"column": 85
} | {
"line": 58,
"column": 0
} | [
{
"pp": "X : Scheme\n⊢ IsGrothendieckAbelian.{u + 1, u + 1, u + 2} (Sheaf (ProEt.topology X) Ab)",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"CategoryTheory.Functor.op",
"AddCommGrpCat.FilteredColimits.forget_preservesFilteredColimits",
"CategoryTheory.GrothendieckTopo... | [] | have : EssentiallySmall.{u + 1} X.ProEt := inferInstance
exact Sheaf.isGrothendieckAbelian_of_essentiallySmall (ProEt.topology X) Ab.{u + 1} | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Padics.PadicIntegers | {
"line": 262,
"column": 2
} | {
"line": 262,
"column": 7
} | {
"line": 263,
"column": 2
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nε : ℝ\nhε : 0 < ε\nk : ℕ\nhk : ε⁻¹ < ↑k\n⊢ ∃ k, ↑p ^ (-↑k) < ε",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Real",
"Real.instDivInvMonoid",
"DivInvMonoid.toZPow",
"Real.instLT",
"Int.instNegInt",
"Int",
... | [
"case h\np : ℕ\nhp : Fact (Nat.Prime p)\nε : ℝ\nhε : 0 < ε\nk : ℕ\nhk : ε⁻¹ < ↑k\n⊢ ↑p ^ (-↑k) < ε"
] | use k | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.NumberTheory.Padics.PadicIntegers | {
"line": 274,
"column": 2
} | {
"line": 274,
"column": 7
} | {
"line": 275,
"column": 2
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nε : ℚ\nhε : 0 < ε\nk : ℕ\nhk : ↑p ^ (-↑k) < ↑ε\n⊢ ∃ k, ↑p ^ (-↑k) < ε",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Rat",
"Int.instNegInt",
"Int",
"Nat.cast",
"Rat.instPowInt",
"HPow.hPow",
"Nat",
"... | [
"case h\np : ℕ\nhp : Fact (Nat.Prime p)\nε : ℚ\nhε : 0 < ε\nk : ℕ\nhk : ↑p ^ (-↑k) < ↑ε\n⊢ ↑p ^ (-↑k) < ε"
] | use k | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.NumberTheory.Padics.PadicIntegers | {
"line": 370,
"column": 4
} | {
"line": 370,
"column": 69
} | {
"line": 371,
"column": 4
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nz : ℤ_[p]\nh : IsUnit z\nw : ℤ_[p]\neq : 1 = z * w\n⊢ 1 ≤ ‖z‖",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"NormedCommRing.toSeminormedCommRing",
"R... | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nz : ℤ_[p]\nh : IsUnit z\nw : ℤ_[p]\neq : 1 = z * w\nthis : ‖z‖ * ‖w‖ ≤ ‖z‖ * 1\n⊢ 1 ≤ ‖z‖"
] | have := mul_le_mul_of_nonneg_left (norm_le_one w) (norm_nonneg z) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.Padics.PadicIntegers | {
"line": 502,
"column": 52
} | {
"line": 504,
"column": 40
} | {
"line": 506,
"column": 0
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\n⊢ ↑p ∈ nonunits ℤ_[p]",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Real.partialOrder",
"Real",
"Nat.Prime",
"Preorder.toLT",
"GroupWithZero.toDivisionMonoid",
"InvOneClass.toOne",
"DivI... | [] | by
have : (p : ℝ)⁻¹ < 1 := inv_lt_one_of_one_lt₀ <| mod_cast hp.out.one_lt
rwa [← norm_p, ← mem_nonunits] at this | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.EpiMono | {
"line": 58,
"column": 4
} | {
"line": 58,
"column": 58
} | {
"line": 59,
"column": 4
} | [
{
"pp": "case mp\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX Y : C\nf : X ⟶ Y\nc : PullbackCone f f\nhc : IsLimit c\nh : c.fst = c.snd\nφ : X ⟶ c.pt\nhφ₁ : φ ≫ c.fst = 𝟙 X\nhφ₂ : φ ≫ c.snd = 𝟙 X\n⊢ IsIso c.fst",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"CategoryTheory.Catego... | [
"case mp.refine_1\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX Y : C\nf : X ⟶ Y\nc : PullbackCone f f\nhc : IsLimit c\nh : c.fst = c.snd\nφ : X ⟶ c.pt\nhφ₁ : φ ≫ c.fst = 𝟙 X\nhφ₂ : φ ≫ c.snd = 𝟙 X\n⊢ (c.fst ≫ φ) ≫ c.fst = 𝟙 c.pt ≫ c.fst",
"case mp.refine_2\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX Y :... | refine ⟨φ, PullbackCone.IsLimit.hom_ext hc ?_ ?_, hφ₁⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.Sites.Point.Conservative | {
"line": 218,
"column": 4
} | {
"line": 218,
"column": 75
} | {
"line": 219,
"column": 4
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nJ : GrothendieckTopology C\nP : ObjectProperty J.Point\ninst✝¹ : LocallySmall.{w, v, u} C\nhP :\n ∀ ⦃X : C⦄ (S : Sieve X),\n (∀ (Φ : P.FullSubcategory) (x : Φ.obj.fiber.obj X),\n ∃ Y g, ∃ (_ : S.arrows g), ∃ y, (ConcreteCategory.hom (Φ.obj.fiber.map g)... | [
"C : Type u\ninst✝² : Category.{v, u} C\nJ : GrothendieckTopology C\nP : ObjectProperty J.Point\ninst✝¹ : LocallySmall.{w, v, u} C\nhP :\n ∀ ⦃X : C⦄ (S : Sieve X),\n (∀ (Φ : P.FullSubcategory) (x : Φ.obj.fiber.obj X),\n ∃ Y g, ∃ (_ : S.arrows g), ∃ y, (ConcreteCategory.hom (Φ.obj.fiber.map g)) y = x) →\n... | have hg := shrinkYoneda_obj_map_shrinkYonedaObjObjEquiv_symm g.op (𝟙 _) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Sites.Point.Conservative | {
"line": 232,
"column": 6
} | {
"line": 234,
"column": 61
} | {
"line": 241,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nJ : GrothendieckTopology C\nP : ObjectProperty J.Point\ninst✝¹ : LocallySmall.{w, v, u} C\nhP :\n ∀ ⦃X : C⦄ (S : Sieve X),\n (∀ (Φ : P.FullSubcategory) (x : Φ.obj.fiber.obj X),\n ∃ Y g, ∃ (_ : S.arrows g), ∃ y, (ConcreteCategory.hom (Φ.obj.fiber.map g)... | [] | rw [← hx₁]
refine Eq.trans (congr_arg _ ht)
(Φ.obj.toPresheafFiber_naturality_apply f _ v y).symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.Point.Conservative | {
"line": 232,
"column": 6
} | {
"line": 234,
"column": 61
} | {
"line": 241,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nJ : GrothendieckTopology C\nP : ObjectProperty J.Point\ninst✝¹ : LocallySmall.{w, v, u} C\nhP :\n ∀ ⦃X : C⦄ (S : Sieve X),\n (∀ (Φ : P.FullSubcategory) (x : Φ.obj.fiber.obj X),\n ∃ Y g, ∃ (_ : S.arrows g), ∃ y, (ConcreteCategory.hom (Φ.obj.fiber.map g)... | [] | rw [← hx₁]
refine Eq.trans (congr_arg _ ht)
(Φ.obj.toPresheafFiber_naturality_apply f _ v y).symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.DoldKan.Compatibility | {
"line": 274,
"column": 2
} | {
"line": 278,
"column": 56
} | {
"line": 280,
"column": 0
} | [
{
"pp": "A : Type u_1\nA' : Type u_2\nB : Type u_3\nB' : Type u_4\ninst✝³ : Category.{v_1, u_1} A\ninst✝² : Category.{v_2, u_2} A'\ninst✝¹ : Category.{v_3, u_3} B\ninst✝ : Category.{v_4, u_4} B'\neA : A ≌ A'\neB : B ≌ B'\ne' : A' ≌ B'\nF : A ⥤ B'\nhF : eA.functor ⋙ e'.functor ≅ F\nG : B ⥤ A\nhG : eB.functor ⋙ e... | [] | ext1; apply NatTrans.ext; ext X
dsimp [equivalence]
simp only [assoc, comp_id, equivalenceUnitIso_hom_app, equivalence₂_inverse, Functor.comp_obj,
id_comp, equivalence₂UnitIso_eq eB hF, equivalence₂UnitIso_hom_app,
← eA.inverse.map_comp_assoc, assoc, ← hε, υ_hom_app] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.DoldKan.Compatibility | {
"line": 274,
"column": 2
} | {
"line": 278,
"column": 56
} | {
"line": 280,
"column": 0
} | [
{
"pp": "A : Type u_1\nA' : Type u_2\nB : Type u_3\nB' : Type u_4\ninst✝³ : Category.{v_1, u_1} A\ninst✝² : Category.{v_2, u_2} A'\ninst✝¹ : Category.{v_3, u_3} B\ninst✝ : Category.{v_4, u_4} B'\neA : A ≌ A'\neB : B ≌ B'\ne' : A' ≌ B'\nF : A ⥤ B'\nhF : eA.functor ⋙ e'.functor ≅ F\nG : B ⥤ A\nhG : eB.functor ⋙ e... | [] | ext1; apply NatTrans.ext; ext X
dsimp [equivalence]
simp only [assoc, comp_id, equivalenceUnitIso_hom_app, equivalence₂_inverse, Functor.comp_obj,
id_comp, equivalence₂UnitIso_eq eB hF, equivalence₂UnitIso_hom_app,
← eA.inverse.map_comp_assoc, assoc, ← hε, υ_hom_app] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.MorphismProperty.Representable | {
"line": 430,
"column": 2
} | {
"line": 430,
"column": 15
} | {
"line": 433,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nF G : Cᵒᵖ ⥤ Type v₁\nf : F ⟶ G\nhf : (monomorphisms C).presheaf f\n⊢ ∀ {X : C} {a b : yoneda.obj X ⟶ F}, a ≫ f = b ≫ f → a = b",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"CategoryTheory.Functor",
"Opposite",
"Categor... | [
"C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nF G : Cᵒᵖ ⥤ Type v₁\nf : F ⟶ G\nhf : (monomorphisms C).presheaf f\nX : C\na b : yoneda.obj X ⟶ F\nh : a ≫ f = b ≫ f\n⊢ a = b"
] | intro X a b h | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.CategoryTheory.MorphismProperty.Representable | {
"line": 434,
"column": 42
} | {
"line": 435,
"column": 61
} | {
"line": 437,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nF G : Cᵒᵖ ⥤ Type v₁\nf : F ⟶ G\nhf : (monomorphisms C).presheaf f\nX : C\na b : yoneda.obj X ⟶ F\nh : a ≫ f = b ≫ f\nthis : ⋯.lift a (𝟙 X) ⋯ = ⋯.lift b (𝟙 X) ⋯\n⊢ a = b",
"ppTerm": "?m.132",
"assigned": true,
"usedConstants": [
"CategoryTheo... | [] | by
simpa using yoneda.congr_map this =≫ (hf.rep.fst (a ≫ f)) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.DoldKan.Projections | {
"line": 70,
"column": 48
} | {
"line": 72,
"column": 6
} | {
"line": 74,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject C\nq : ℕ\n⊢ P q + Q q = 𝟙 K[X]",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ChainComplex",
"HomologicalComplex.instCategory",
"Nat.instOne",
"Cate... | [] | by
rw [Q]
abel | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 1010,
"column": 4
} | {
"line": 1010,
"column": 21
} | {
"line": 1011,
"column": 4
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nf : CauSeq ℚ_[p] norm\ncau_seq_norm_e : IsCauSeq ⇑padicNormE ↑f\nq : ℚ_[p]\nhq : ∀ ε > 0, ∃ N, ∀ i ≥ N, padicNormE (↑⟨↑f, cau_seq_norm_e⟩ i - q) < ε\nε : ℝ\nhε : ε > 0\nε' : ℚ\nhε' : 0 < ε' ∧ ↑ε' < ε\nN : ℕ\nhN : ∀ i ≥ N, padicNormE (↑⟨↑f, cau_seq_norm_e⟩ i - q) < ε'\ni ... | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nf : CauSeq ℚ_[p] norm\ncau_seq_norm_e : IsCauSeq ⇑padicNormE ↑f\nq : ℚ_[p]\nhq : ∀ ε > 0, ∃ N, ∀ i ≥ N, padicNormE (↑⟨↑f, cau_seq_norm_e⟩ i - q) < ε\nε : ℝ\nhε : ε > 0\nε' : ℚ\nhε' : 0 < ε' ∧ ↑ε' < ε\nN : ℕ\nhN : ∀ i ≥ N, padicNormE (↑⟨↑f, cau_seq_norm_e⟩ i - q) < ε'\ni : ℕ\nhi : i ... | have h := hN i hi | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.AlgebraicTopology.DoldKan.Decomposition | {
"line": 133,
"column": 2
} | {
"line": 133,
"column": 39
} | {
"line": 135,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject C\nn : ℕ\nZ Z' : C\nf : MorphComponents X n Z\nh : Z ⟶ Z'\n⊢ PInfty.f (n + 1) ≫ { a := f.a ≫ h, b := fun i ↦ f.b i ≫ h }.a +\n ∑ i, (P ↑i).f (n + 1) ≫ X.δ i.rev.succ ≫ { a := f.a ≫ h, b := fun i ↦ f.b i ≫ h }... | [] | simp only [add_comp, sum_comp, assoc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicTopology.SimplicialObject.Split | {
"line": 163,
"column": 4
} | {
"line": 163,
"column": 28
} | {
"line": 165,
"column": 0
} | [
{
"pp": "case mpr\nΔ : SimplexCategoryᵒᵖ\nA : IndexSet Δ\na✝ : Mono A.e\n⊢ (unop Δ).len ≤ (unop A.fst).len",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Opposite",
"CategoryTheory.Epi",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"SimplexCategory... | [] | exact len_le_of_mono A.e | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicTopology.DoldKan.Faces | {
"line": 197,
"column": 8
} | {
"line": 197,
"column": 30
} | {
"line": 197,
"column": 30
} | [
{
"pp": "case neg.inr\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nq a m : ℕ\nφ : Y ⟶ X _⦋m + 1 + 1⦌\nv : HigherFacesVanish q φ\nj : Fin (m + 1 + 1)\nhj₁ : m + 1 + 1 ≤ ↑j + (q + 1)\nhqn : q ≤ m + 1\nha : q + a = m + 1\nhj₂ : ¬a = ↑j\nhaj : a < ↑j\nham : a ... | [
"case neg.inr\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nq a m : ℕ\nφ : Y ⟶ X _⦋m + 1 + 1⦌\nv : HigherFacesVanish q φ\nj : Fin (m + 1 + 1)\nhj₁ : m + 1 + 1 ≤ ↑j + (q + 1)\nhqn : q ≤ m + 1\nha : q + a = m + 1\nhj₂ : ¬a = ↑j\nhaj : a < ↑j\nham : a ≤ m\nham'' :... | X.δ_comp_δ_self'_assoc | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.DoldKan.Degeneracies | {
"line": 132,
"column": 2
} | {
"line": 137,
"column": 7
} | {
"line": 138,
"column": 2
} | [
{
"pp": "case zero\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject C\nΔ' : SimplexCategory\nθ : ⦋0⦌ ⟶ Δ'\nhθ : ¬Function.Injective ⇑(SimplexCategory.Hom.toOrderHom θ)\n⊢ X.map θ.op ≫ PInfty.f 0 = 0",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
... | [
"case succ\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject C\nΔ' : SimplexCategory\nn✝ : ℕ\nθ : ⦋n✝ + 1⦌ ⟶ Δ'\nhθ : ¬Function.Injective ⇑(SimplexCategory.Hom.toOrderHom θ)\n⊢ X.map θ.op ≫ PInfty.f (n✝ + 1) = 0"
] | · exfalso
apply hθ
intro x y h
fin_cases x
fin_cases y
rfl | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicTopology.DoldKan.GammaCompN | {
"line": 51,
"column": 15
} | {
"line": 51,
"column": 50
} | {
"line": 51,
"column": 51
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\ninst✝ : HasFiniteCoproducts C\nK : ChainComplex C ℕ\nn : ℕ\n⊢ K.d (n + 1) n =\n ((Γ₀.splitting K).cofan (op ⦋n + 1⦌)).inj (Splitting.IndexSet.id (op ⦋n + 1⦌)) ≫\n (Γ₀.obj K).map (SimplexCategory.δ 0).op ≫ (Γ₀.splitting K).πSu... | [
"C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\ninst✝ : HasFiniteCoproducts C\nK : ChainComplex C ℕ\nn : ℕ\n⊢ K.d (n + 1) n =\n Γ₀.Obj.Termwise.mapMono K (SimplexCategory.δ 0) ≫\n ((Γ₀.splitting K).cofan (op ⦋n⦌)).inj (Splitting.IndexSet.id (op ⦋n⦌)) ≫\n (Γ₀.splitting K).πSumma... | Γ₀.Obj.mapMono_on_summand_id_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.DoldKan.GammaCompN | {
"line": 58,
"column": 15
} | {
"line": 58,
"column": 50
} | {
"line": 58,
"column": 51
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\ninst✝ : HasFiniteCoproducts C\nK : ChainComplex C ℕ\nn : ℕ\ni : Fin (n + 2)\nhi : i ≠ 0\n⊢ (-1) ^ ↑i •\n ((Γ₀.splitting K).cofan (op ⦋n + 1⦌)).inj (Splitting.IndexSet.id (op ⦋n + 1⦌)) ≫\n (Γ₀.obj K).map (SimplexCategory.δ... | [
"C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\ninst✝ : HasFiniteCoproducts C\nK : ChainComplex C ℕ\nn : ℕ\ni : Fin (n + 2)\nhi : i ≠ 0\n⊢ (-1) ^ ↑i •\n Γ₀.Obj.Termwise.mapMono K (SimplexCategory.δ i) ≫\n ((Γ₀.splitting K).cofan (op ⦋n⦌)).inj (Splitting.IndexSet.id (op ⦋n⦌)) ≫\n ... | Γ₀.Obj.mapMono_on_summand_id_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.DoldKan.GammaCompN | {
"line": 134,
"column": 2
} | {
"line": 134,
"column": 26
} | {
"line": 135,
"column": 2
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\ninst✝ : HasFiniteCoproducts C\nX : ChainComplex C ℕ\nn : ℕ\n⊢ (N₂Γ₂ToKaroubiIso.hom.app X).f.f n = PInfty.f n",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"CategoryTheory.Idempotents.Karoubi.Hom.f",
... | [
"C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\ninst✝ : HasFiniteCoproducts C\nX : ChainComplex C ℕ\nn : ℕ\n⊢ (((PInfty.f n ≫\n (Γ₀.splitting X).desc (op ⦋n⦌) fun A ↦ 𝟙 (X.X (unop A.fst).len) ≫ ((Γ₀.splitting X).cofan (op ⦋n⦌)).inj A) ≫\n PInfty.f n ≫ 𝟙 (Γ₀.Obj.obj₂ X (o... | dsimp [N₂Γ₂ToKaroubiIso] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.AlgebraicTopology.DoldKan.GammaCompN | {
"line": 150,
"column": 2
} | {
"line": 150,
"column": 26
} | {
"line": 151,
"column": 2
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\ninst✝ : HasFiniteCoproducts C\nX : ChainComplex C ℕ\nn : ℕ\n⊢ (N₂Γ₂ToKaroubiIso.inv.app X).f.f n = PInfty.f n",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"CategoryTheory.Idempotents.Karoubi.Hom.f",
... | [
"C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\ninst✝ : HasFiniteCoproducts C\nX : ChainComplex C ℕ\nn : ℕ\n⊢ (PInfty.f n ≫\n (PInfty.f n ≫ 𝟙 (Γ₀.Obj.obj₂ X (op ⦋n⦌))) ≫\n (PInfty.f n ≫ 𝟙 (Γ₀.Obj.obj₂ X (op ⦋n⦌))) ≫\n PInfty.f n ≫\n (Γ₀.splitting X).desc (op ... | dsimp [N₂Γ₂ToKaroubiIso] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.CategoryTheory.Localization.CalculusOfFractions.OfAdjunction | {
"line": 74,
"column": 26
} | {
"line": 86,
"column": 85
} | {
"line": 88,
"column": 0
} | [
{
"pp": "C₁ : Type u_1\nC₂ : Type u_2\ninst✝³ : Category.{v_1, u_1} C₁\ninst✝² : Category.{v_2, u_2} C₂\nG : C₁ ⥤ C₂\nF : C₂ ⥤ C₁\ninst✝¹ : F.Full\ninst✝ : F.Faithful\nadj : G ⊣ F\nW : MorphismProperty C₁\nhW : W.IsInvertedBy G\nhW' : W.functorCategory C₁ adj.unit\n⊢ G.IsLocalization W",
"ppTerm": "?m.37",
... | [] | by
let Φ : W.Localization ⥤ C₂ := Localization.lift _ hW W.Q
let e : W.Q ⋙ Φ ≅ G := by apply Localization.fac
have : IsIso (Functor.whiskerRight adj.unit W.Q) := by
rw [NatTrans.isIso_iff_isIso_app]
intro X
exact Localization.inverts W.Q W _ (hW' X)
exact Functor.IsLocalization.of_equivalence_target... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.ModelCategory.CofibrantObjectHomotopy | {
"line": 311,
"column": 2
} | {
"line": 317,
"column": 61
} | {
"line": 318,
"column": 2
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : ModelCategory C\nHcof : Type u_1 := (weakEquivalences (HoCat C)).Localization\nLcofπ : HoCat C ⥤ Hcof := (weakEquivalences (HoCat C)).Q\nLcof : CofibrantObject C ⥤ Hcof := toHoCat ⋙ Lcofπ\nH : Type u_1 := (weakEquivalences C).Localization\nL : C ⥤ H... | [
"C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : ModelCategory C\nHcof : Type u_1 := (weakEquivalences (HoCat C)).Localization\nLcofπ : HoCat C ⥤ Hcof := (weakEquivalences (HoCat C)).Q\nLcof : CofibrantObject C ⥤ Hcof := toHoCat ⋙ Lcofπ\nH : Type u_1 := (weakEquivalences C).Localization\nL : C ⥤ H := (weakEqu... | let E : Hcof ≌ H := CategoryTheory.Equivalence.mk F G
(Localization.liftNatIso Lcof (weakEquivalences _) Lcof (ι ⋙ HoCat.resolution ⋙ Lcofπ) _ _
((asIso (whiskerRight HoCat.ιCompResolutionNatTrans Lcofπ)).symm ≪≫
associator _ _ _))
(Localization.liftNatIso L (weakEquivalences _)
(HoCat.res... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.CategoryTheory.Limits.Set | {
"line": 43,
"column": 6
} | {
"line": 43,
"column": 90
} | {
"line": 45,
"column": 0
} | [
{
"pp": "case hinj\nJ : Type w\ninst✝¹ : Category.{w', w} J\nX : Type u\ninst✝ : IsFilteredOrEmpty J\nF : J ⥤ Set X\ni j : J\nx : X\nhx : x ∈ F.obj i\nhy : x ∈ F.obj j\nh :\n (ConcreteCategory.hom ((functorToTypes.mapCocone (colimitCocone F).cocone).ι.app i)) ⟨x, hx⟩ =\n (ConcreteCategory.hom ((functorToTyp... | [] | exact ⟨IsFiltered.max i j, IsFiltered.leftToMax i j, IsFiltered.rightToMax i j, rfl⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Limits.Types.Pushouts | {
"line": 268,
"column": 23
} | {
"line": 268,
"column": 35
} | {
"line": 268,
"column": 35
} | [
{
"pp": "X₁ X₂ X₃ X₄ : Type u\nt : X₁ ⟶ X₂\nr : X₂ ⟶ X₄\nl : X₁ ⟶ X₃\nb : X₃ ⟶ X₄\nh : IsPushout t l r b\nx : (span t l).obj none\n⊢ (ConcreteCategory.hom r) ((ConcreteCategory.hom t) x) = (ConcreteCategory.hom (h.cocone.ι.app none)) x",
"ppTerm": "?m.95",
"assigned": true,
"usedConstants": [
... | [] | by simp; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Category.ReflQuiv | {
"line": 200,
"column": 41
} | {
"line": 203,
"column": 24
} | {
"line": 205,
"column": 0
} | [
{
"pp": "V : Type u_1\ninst✝ : ReflQuiver V\nx y : V\ne : x ⟶ y\n⊢ morphismPropertyHomMk V (homMk e)",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"CategoryTheory.MorphismProperty.ofHoms_iff",
"Eq.mpr",
"CategoryTheory.Cat.FreeRefl.mk",
"CategoryTheory.Cat.FreeRef... | [] | by
dsimp only [morphismPropertyHomMk]
rw [MorphismProperty.ofHoms_iff]
exact ⟨⟨x, y, e⟩, rfl⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Category.ReflQuiv | {
"line": 246,
"column": 4
} | {
"line": 247,
"column": 8
} | {
"line": 247,
"column": 8
} | [
{
"pp": "V : Type u_1\ninst✝¹ : ReflQuiver V\nD : Type u_2\ninst✝ : Category.{v_1, u_2} D\nF : V ⥤rq D\n⊢ ∀ (x y : Paths V) (f₁ f₂ : x ⟶ y),\n FreeReflRel V x y f₁ f₂ → (Paths.lift F.toPrefunctor).map f₁ = (Paths.lift F.toPrefunctor).map f₂",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": ... | [] | rintro _ _ _ _ ⟨h⟩
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Category.ReflQuiv | {
"line": 246,
"column": 4
} | {
"line": 247,
"column": 8
} | {
"line": 247,
"column": 8
} | [
{
"pp": "V : Type u_1\ninst✝¹ : ReflQuiver V\nD : Type u_2\ninst✝ : Category.{v_1, u_2} D\nF : V ⥤rq D\n⊢ ∀ (x y : Paths V) (f₁ f₂ : x ⟶ y),\n FreeReflRel V x y f₁ f₂ → (Paths.lift F.toPrefunctor).map f₁ = (Paths.lift F.toPrefunctor).map f₂",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": ... | [] | rintro _ _ _ _ ⟨h⟩
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SimplicialSet.HomotopyCat | {
"line": 285,
"column": 45
} | {
"line": 285,
"column": 57
} | {
"line": 285,
"column": 58
} | [
{
"pp": "V : Truncated 2\nx₀ x₁ : V.obj (op { obj := ⦋0⦌, property := OneTruncation₂._proof_1 })\ne : Edge x₀ x₁\ny₀ y₁ : V.obj (op { obj := ⦋0⦌, property := OneTruncation₂._proof_1 })\ne' : Edge y₀ y₁\nh : e.edge = e'.edge\n⊢ (ConcreteCategory.hom (V.map (δ₂ 1 Edge._proof_1 Edge._proof_3).op)) e.edge = y₀",
... | [
"V : Truncated 2\nx₀ x₁ : V.obj (op { obj := ⦋0⦌, property := OneTruncation₂._proof_1 })\ne : Edge x₀ x₁\ny₀ y₁ : V.obj (op { obj := ⦋0⦌, property := OneTruncation₂._proof_1 })\ne' : Edge y₀ y₁\nh : e.edge = e'.edge\n⊢ (ConcreteCategory.hom (V.map (δ₂ 1 Edge._proof_1 Edge._proof_3).op)) e.edge =\n (ConcreteCateg... | ← e'.src_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.Reedy.Basic | {
"line": 134,
"column": 2
} | {
"line": 134,
"column": 45
} | {
"line": 135,
"column": 2
} | [
{
"pp": "C : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\nW₁ W₂ : MorphismProperty C\ninst✝⁵ : W₁.IsMultiplicative\ninst✝⁴ : W₂.IsMultiplicative\nα : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : OrderBot α\ninst✝¹ : SuccOrder α\ninst✝ : WellFoundedLT α\nr : ReedyStructure W₁ W₂ α\nX Y : C\nf : X ⟶ Y\nh : W₁.MapFacto... | [
"C : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\nW₁ W₂ : MorphismProperty C\ninst✝⁵ : W₁.IsMultiplicative\ninst✝⁴ : W₂.IsMultiplicative\nα : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : OrderBot α\ninst✝¹ : SuccOrder α\ninst✝ : WellFoundedLT α\nr : ReedyStructure W₁ W₂ α\nX Y : C\nf : X ⟶ Y\nh : W₁.MapFactorizationData... | have := r.subsingleton_mapFactorizationData | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.AlgebraicTopology.Reedy.Basic | {
"line": 144,
"column": 34
} | {
"line": 152,
"column": 41
} | {
"line": 154,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\nW₁ W₂ : MorphismProperty C\ninst✝⁵ : W₁.IsMultiplicative\ninst✝⁴ : W₂.IsMultiplicative\nα : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : OrderBot α\ninst✝¹ : SuccOrder α\ninst✝ : WellFoundedLT α\nr : ReedyStructure W₁ W₂ α\nX Z Y : C\nf : X ⟶ Z\ng : Z ⟶ Y\n⊢ ... | [] | by
obtain ⟨Zf, f₁, f₂, hf₁, hf₂, fac_f, eq_f⟩ := r.exists_fac f
obtain ⟨Zg, g₁, g₂, hg₁, hg₂, fac_g, eq_g⟩ := r.exists_fac g
obtain ⟨Zh, h₁, h₂, hh₁, hh₂, fac_h, eq_h⟩ := r.exists_fac (f₂ ≫ g₁)
let factfg := MorphismProperty.MapFactorizationData.mk (f := f ≫ g) Zh (f₁ ≫ h₁) (h₂ ≫ g₂)
(by simp [reassoc_of% f... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.SimplicialSet.Coskeletal | {
"line": 114,
"column": 67
} | {
"line": 114,
"column": 79
} | {
"line": 114,
"column": 79
} | [
{
"pp": "X : SSet\nsx : X.StrictSegal\nn : ℕ\ns : Cone (proj (op ⦋n⦌) (inclusion 2).op ⋙ (inclusion 2).op ⋙ X)\nx : s.pt\ni : ℕ\nhij : i ≤ i\nhj : i ≤ n\nthis : mkOfLe ⟨i, ⋯⟩ ⟨i, ⋯⟩ ⋯ = ⦋1⦌.const ⦋0⦌ 0 ≫ ⦋0⦌.const ⦋n⦌ ⟨i, ⋯⟩\n⊢ (strArrowMk₂ (⦋0⦌.const ⦋n⦌ ⟨i, ⋯⟩) lift._proof_1).hom ≫ (inclusion 2).op.map (Hom.t... | [] | by simp; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.SimplicialSet.HomotopyCat | {
"line": 313,
"column": 41
} | {
"line": 316,
"column": 24
} | {
"line": 318,
"column": 0
} | [
{
"pp": "V : Truncated 2\nx y : V.obj (op { obj := ⦋0⦌, property := OneTruncation₂._proof_1 })\ne : Edge x y\n⊢ morphismPropertyHomMk V (homMk e)",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"CategoryTheory.MorphismProperty.ofHoms_iff",
"SSet.Truncated.Edge",
"Eq.mpr",... | [] | by
dsimp only [morphismPropertyHomMk]
rw [MorphismProperty.ofHoms_iff]
exact ⟨⟨x, y, e⟩, rfl⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.Reedy.Basic | {
"line": 170,
"column": 2
} | {
"line": 172,
"column": 31
} | {
"line": 174,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\nW₁ W₂ : MorphismProperty C\ninst✝⁵ : W₁.IsMultiplicative\ninst✝⁴ : W₂.IsMultiplicative\nα : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : OrderBot α\ninst✝¹ : SuccOrder α\ninst✝ : WellFoundedLT α\nr : ReedyStructure W₁ W₂ α\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\n⊢ ... | [] | have ⟨_, g₁, g₂, _, _, h_fac, h_deg⟩ := r.exists_fac g
rw [h_deg, ← h_fac, <- Category.assoc]
exact r.degHom_le (f ≫ g₁) g₂ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.Reedy.Basic | {
"line": 170,
"column": 2
} | {
"line": 172,
"column": 31
} | {
"line": 174,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C\nW₁ W₂ : MorphismProperty C\ninst✝⁵ : W₁.IsMultiplicative\ninst✝⁴ : W₂.IsMultiplicative\nα : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : OrderBot α\ninst✝¹ : SuccOrder α\ninst✝ : WellFoundedLT α\nr : ReedyStructure W₁ W₂ α\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\n⊢ ... | [] | have ⟨_, g₁, g₂, _, _, h_fac, h_deg⟩ := r.exists_fac g
rw [h_deg, ← h_fac, <- Category.assoc]
exact r.degHom_le (f ≫ g₁) g₂ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SimplexCategory.GeneratorsRelations.Basic | {
"line": 141,
"column": 50
} | {
"line": 141,
"column": 60
} | {
"line": 141,
"column": 60
} | [
{
"pp": "P : MorphismProperty SimplexCategoryGenRel\nid : ∀ {n : ℕ}, P (𝟙 (mk n))\ncomp_δ : ∀ {n m : ℕ} (u : mk n ⟶ mk m) (i : Fin (m + 2)), P u → P (u ≫ δ i)\ncomp_σ : ∀ {n m : ℕ} (u : mk n ⟶ mk (m + 1)) (i : Fin (m + 1)), P u → P (u ≫ σ i)\na b : SimplexCategoryGenRel\nf : a ⟶ b\nthis : ⊤ ≤ P\n⊢ P f",
"p... | [
"P : MorphismProperty SimplexCategoryGenRel\nid : ∀ {n : ℕ}, P (𝟙 (mk n))\ncomp_δ : ∀ {n m : ℕ} (u : mk n ⟶ mk m) (i : Fin (m + 2)), P u → P (u ≫ δ i)\ncomp_σ : ∀ {n m : ℕ} (u : mk n ⟶ mk (m + 1)) (i : Fin (m + 1)), P u → P (u ≫ σ i)\na b : SimplexCategoryGenRel\nf : a ⟶ b\nthis : P = ⊤\n⊢ P f"
] | top_le_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.SimplicialSet.Coskeletal | {
"line": 136,
"column": 6
} | {
"line": 138,
"column": 33
} | {
"line": 139,
"column": 6
} | [
{
"pp": "case succ\nX : SSet\nsx : X.StrictSegal\nn : ℕ\ns : Cone (proj (op ⦋n⦌) (inclusion 2).op ⋙ (inclusion 2).op ⋙ X)\nx : s.pt\nk : ℕ\nhk :\n ∀ (i j : ℕ) (hij : i ≤ j) (hj : j ≤ n),\n i + k = j →\n (ConcreteCategory.hom (X.map (mkOfLe ⟨i, ⋯⟩ ⟨j, ⋯⟩ hij).op)) (lift sx s x) =\n (ConcreteCateg... | [
"case succ\nX : SSet\nsx : X.StrictSegal\nn : ℕ\ns : Cone (proj (op ⦋n⦌) (inclusion 2).op ⋙ (inclusion 2).op ⋙ X)\nx : s.pt\nk : ℕ\nhk :\n ∀ (i j : ℕ) (hij : i ≤ j) (hj : j ≤ n),\n i + k = j →\n (ConcreteCategory.hom (X.map (mkOfLe ⟨i, ⋯⟩ ⟨j, ⋯⟩ hij).op)) (lift sx s x) =\n (ConcreteCategory.hom (s.π... | have h₀ : X.map α₀.hom (lift sx s x) = s.π.app α₀ x := by
subst hik
exact fac_aux₁ _ _ _ _ hj | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.AlgebraicTopology.SimplexCategory.GeneratorsRelations.Basic | {
"line": 167,
"column": 48
} | {
"line": 167,
"column": 58
} | {
"line": 167,
"column": 58
} | [
{
"pp": "P : MorphismProperty SimplexCategoryGenRel\nid : ∀ {n : ℕ}, P (𝟙 (mk n))\nδ_comp : ∀ {n m : ℕ} (u : mk (m + 1) ⟶ mk n) (i : Fin (m + 2)), P u → P (δ i ≫ u)\nσ_comp : ∀ {n m : ℕ} (u : mk m ⟶ mk n) (i : Fin (m + 1)), P u → P (σ i ≫ u)\na b : SimplexCategoryGenRel\nf : a ⟶ b\nthis : ⊤ ≤ P\n⊢ P f",
"p... | [
"P : MorphismProperty SimplexCategoryGenRel\nid : ∀ {n : ℕ}, P (𝟙 (mk n))\nδ_comp : ∀ {n m : ℕ} (u : mk (m + 1) ⟶ mk n) (i : Fin (m + 2)), P u → P (δ i ≫ u)\nσ_comp : ∀ {n m : ℕ} (u : mk m ⟶ mk n) (i : Fin (m + 1)), P u → P (σ i ≫ u)\na b : SimplexCategoryGenRel\nf : a ⟶ b\nthis : P = ⊤\n⊢ P f"
] | top_le_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.SimplexCategory.GeneratorsRelations.Basic | {
"line": 218,
"column": 2
} | {
"line": 219,
"column": 46
} | {
"line": 221,
"column": 0
} | [
{
"pp": "n : ℕ\ni : Fin (n + 1)\n⊢ δ i.castSucc ≫ σ i = 𝟙 (mk n)",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"FreeSimplexQuiver.homRel",
"CategoryTheory.CategoryStruct.toQuiver",
"SimplexCategoryGenRel.mk",
"CategoryTheory.Paths.categoryPaths",
"CategoryT... | [] | apply CategoryTheory.Quotient.sound
exact FreeSimplexQuiver.homRel.δ_comp_σ_self | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplexCategory.GeneratorsRelations.Basic | {
"line": 218,
"column": 2
} | {
"line": 219,
"column": 46
} | {
"line": 221,
"column": 0
} | [
{
"pp": "n : ℕ\ni : Fin (n + 1)\n⊢ δ i.castSucc ≫ σ i = 𝟙 (mk n)",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"FreeSimplexQuiver.homRel",
"CategoryTheory.CategoryStruct.toQuiver",
"SimplexCategoryGenRel.mk",
"CategoryTheory.Paths.categoryPaths",
"CategoryT... | [] | apply CategoryTheory.Quotient.sound
exact FreeSimplexQuiver.homRel.δ_comp_σ_self | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SimplicialSet.Coskeletal | {
"line": 123,
"column": 6
} | {
"line": 166,
"column": 11
} | {
"line": 168,
"column": 0
} | [
{
"pp": "case succ\nX : SSet\nsx : X.StrictSegal\nn : ℕ\ns : Cone (proj (op ⦋n⦌) (inclusion 2).op ⋙ (inclusion 2).op ⋙ X)\nx : s.pt\nk : ℕ\nhk :\n ∀ (i j : ℕ) (hij : i ≤ j) (hj : j ≤ n),\n i + k = j →\n (ConcreteCategory.hom (X.map (mkOfLe ⟨i, ⋯⟩ ⟨j, ⋯⟩ hij).op)) (lift sx s x) =\n (ConcreteCateg... | [] | intro i j hij hj hik
let α := strArrowMk₂ (mkOfLeComp (n := n) ⟨i, by omega⟩ ⟨i + k, by omega⟩
⟨j, by omega⟩ (by simp) (by simp only [Fin.mk_le_mk]; omega))
let α₀ := strArrowMk₂ (mkOfLe (n := n) ⟨i + k, by omega⟩ ⟨j, by omega⟩
(by simp only [Fin.mk_le_mk]; omega))
let α₁ := strArrow... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplicialSet.Coskeletal | {
"line": 123,
"column": 6
} | {
"line": 166,
"column": 11
} | {
"line": 168,
"column": 0
} | [
{
"pp": "case succ\nX : SSet\nsx : X.StrictSegal\nn : ℕ\ns : Cone (proj (op ⦋n⦌) (inclusion 2).op ⋙ (inclusion 2).op ⋙ X)\nx : s.pt\nk : ℕ\nhk :\n ∀ (i j : ℕ) (hij : i ≤ j) (hj : j ≤ n),\n i + k = j →\n (ConcreteCategory.hom (X.map (mkOfLe ⟨i, ⋯⟩ ⟨j, ⋯⟩ hij).op)) (lift sx s x) =\n (ConcreteCateg... | [] | intro i j hij hj hik
let α := strArrowMk₂ (mkOfLeComp (n := n) ⟨i, by omega⟩ ⟨i + k, by omega⟩
⟨j, by omega⟩ (by simp) (by simp only [Fin.mk_le_mk]; omega))
let α₀ := strArrowMk₂ (mkOfLe (n := n) ⟨i + k, by omega⟩ ⟨j, by omega⟩
(by simp only [Fin.mk_le_mk]; omega))
let α₁ := strArrow... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SimplicialSet.HoFunctorMonoidal | {
"line": 185,
"column": 26
} | {
"line": 185,
"column": 43
} | {
"line": 185,
"column": 44
} | [
{
"pp": "X X' Y Y' Z : Truncated 2\nx₀ : X.obj (Opposite.op { obj := ⦋0⦌, property := OneTruncation₂._proof_1 })\ny₀ : Y.obj (Opposite.op { obj := ⦋0⦌, property := OneTruncation₂._proof_1 })\nx₁ : X.obj (Opposite.op { obj := ⦋0⦌, property := OneTruncation₂._proof_1 })\ny₁ : Y.obj (Opposite.op { obj := ⦋0⦌, prop... | [
"X X' Y Y' Z : Truncated 2\nx₀ : X.obj (Opposite.op { obj := ⦋0⦌, property := OneTruncation₂._proof_1 })\ny₀ : Y.obj (Opposite.op { obj := ⦋0⦌, property := OneTruncation₂._proof_1 })\nx₁ : X.obj (Opposite.op { obj := ⦋0⦌, property := OneTruncation₂._proof_1 })\ny₁ : Y.obj (Opposite.op { obj := ⦋0⦌, property := OneT... | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.SimplicialSet.Coskeletal | {
"line": 198,
"column": 72
} | {
"line": 198,
"column": 84
} | {
"line": 198,
"column": 84
} | [
{
"pp": "X : SSet\nsx : X.StrictSegal\nn : ℕ\ns : Cone (proj (op ⦋n⦌) (inclusion 2).op ⋙ (inclusion 2).op ⋙ X)\nx : s.pt\ni : ℕ\nhi : i ≤ 2\nf : unop ((inclusion 2).op.obj (op { obj := ⦋i⦌, property := hi })) ⟶ unop (op ⦋n⦌)\nk : Fin (i + 1)\n⊢ (strArrowMk₂ f hi).hom ≫ (inclusion 2).op.map (Hom.tr (⦋0⦌.const ⦋i... | [] | by simp; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.SimplicialSet.HoFunctorMonoidal | {
"line": 341,
"column": 30
} | {
"line": 341,
"column": 47
} | {
"line": 341,
"column": 48
} | [
{
"pp": "X X' Y Y' Z : Truncated 2\nx : X.obj (Opposite.op { obj := ⦋0⦌, property := OneTruncation₂._proof_1 })\ny : Y.obj (Opposite.op { obj := ⦋0⦌, property := OneTruncation₂._proof_1 })\nz₀ z₁ : Z.obj (Opposite.op { obj := ⦋0⦌, property := OneTruncation₂._proof_1 })\ne : Edge z₀ z₁\n⊢ (mapHomotopyCategory (α... | [
"X X' Y Y' Z : Truncated 2\nx : X.obj (Opposite.op { obj := ⦋0⦌, property := OneTruncation₂._proof_1 })\ny : Y.obj (Opposite.op { obj := ⦋0⦌, property := OneTruncation₂._proof_1 })\nz₀ z₁ : Z.obj (Opposite.op { obj := ⦋0⦌, property := OneTruncation₂._proof_1 })\ne : Edge z₀ z₁\n⊢ (mapHomotopyCategory (α_ X Y Z).hom... | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.SimplicialSet.HoFunctorMonoidal | {
"line": 349,
"column": 30
} | {
"line": 349,
"column": 47
} | {
"line": 350,
"column": 10
} | [
{
"pp": "X X' Y Y' Z : Truncated 2\nx : X.obj (Opposite.op { obj := ⦋0⦌, property := OneTruncation₂._proof_1 })\ny₀ y₁ : Y.obj (Opposite.op { obj := ⦋0⦌, property := OneTruncation₂._proof_1 })\ne : Edge y₀ y₁\nz : Z.obj (Opposite.op { obj := ⦋0⦌, property := OneTruncation₂._proof_1 })\n⊢ homMk ((Edge.id x).tens... | [
"X X' Y Y' Z : Truncated 2\nx : X.obj (Opposite.op { obj := ⦋0⦌, property := OneTruncation₂._proof_1 })\ny₀ y₁ : Y.obj (Opposite.op { obj := ⦋0⦌, property := OneTruncation₂._proof_1 })\ne : Edge y₀ y₁\nz : Z.obj (Opposite.op { obj := ⦋0⦌, property := OneTruncation₂._proof_1 })\n⊢ homMk ((Edge.id x).tensor (e.tensor... | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.SimplexCategory.SemiSimplexCategory | {
"line": 82,
"column": 4
} | {
"line": 83,
"column": 38
} | {
"line": 85,
"column": 0
} | [
{
"pp": "X✝ Y✝ : SemiSimplexCategory\na₁✝ a₂✝ : X✝ ⟶ Y✝\nh : toSimplexCategory.map a₁✝ = toSimplexCategory.map a₂✝\n⊢ a₁✝ = a₂✝",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Equiv.instEquivLike",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"SemiSimple... | [] | ext : 2
apply ConcreteCategory.congr_hom h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplexCategory.SemiSimplexCategory | {
"line": 82,
"column": 4
} | {
"line": 83,
"column": 38
} | {
"line": 85,
"column": 0
} | [
{
"pp": "X✝ Y✝ : SemiSimplexCategory\na₁✝ a₂✝ : X✝ ⟶ Y✝\nh : toSimplexCategory.map a₁✝ = toSimplexCategory.map a₂✝\n⊢ a₁✝ = a₂✝",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Equiv.instEquivLike",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"SemiSimple... | [] | ext : 2
apply ConcreteCategory.congr_hom h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SimplicialSet.HoFunctorMonoidal | {
"line": 363,
"column": 6
} | {
"line": 363,
"column": 23
} | {
"line": 363,
"column": 24
} | [
{
"pp": "X Y Z : Truncated 2\nxyz : X.HomotopyCategory × Y.HomotopyCategory × Z.HomotopyCategory\n⊢ 𝟙\n (((prod.associativity X.HomotopyCategory Y.HomotopyCategory Z.HomotopyCategory).inverse ⋙\n (inverse X Y).prod (𝟭 Z.HomotopyCategory) ⋙ inverse (X ⊗ Y) Z ⋙ mapHomotopyCategory (α_ X Y Z)... | [
"X Y Z : Truncated 2\nxyz : X.HomotopyCategory × Y.HomotopyCategory × Z.HomotopyCategory\n⊢ (Functor.currying₃.functor.map\n (mkNatIso\n (fun x ↦\n mkNatIso\n (fun y ↦\n mkNatIso\n (fun z ↦\n ... | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.SimplicialSet.HoFunctorMonoidal | {
"line": 385,
"column": 37
} | {
"line": 385,
"column": 54
} | {
"line": 385,
"column": 55
} | [
{
"pp": "X Y Z : Truncated 2\nxyz : (X.HomotopyCategory × Y.HomotopyCategory) × Z.HomotopyCategory\n⊢ 𝟙 ((mapHomotopyCategory (α_ X Y Z).hom).obj ((inverse (X ⊗ Y) Z).obj ((inverse X Y).obj xyz.1, xyz.2))) ≫\n (mapHomotopyCategory (α_ X Y Z).hom).map\n ((inverse (X ⊗ Y) Z).map (Prod.mkHom (𝟙 ((i... | [
"X Y Z : Truncated 2\nxyz : (X.HomotopyCategory × Y.HomotopyCategory) × Z.HomotopyCategory\n⊢ (mapHomotopyCategory (α_ X Y Z).hom).map\n ((inverse (X ⊗ Y) Z).map (Prod.mkHom (𝟙 ((inverse X Y).obj xyz.1)) (𝟙 xyz.2))) ≫\n 𝟙\n ((mapHomotopyCategory (α_ X Y Z).hom).obj\n ((inverse (X ... | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.SimplicialObject.II | {
"line": 95,
"column": 4
} | {
"line": 95,
"column": 61
} | {
"line": 96,
"column": 2
} | [
{
"pp": "case mp\nn m : ℕ\nf : Fin (n + 1) →o Fin (m + 1)\nx : Fin (m + 2)\ny : Fin (n + 1)\nh : x ≤ (f y).castSucc\nh' : ∀ b ∈ finset f x, y.castSucc ≤ b\ni : Fin (n + 1)\nhi : i < y\nthis : x ≤ (f i).castSucc\n⊢ False",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"S... | [] | exact hi.not_ge (by simpa using h' i.castSucc (by simpa)) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicTopology.SimplicialNerve | {
"line": 126,
"column": 69
} | {
"line": 126,
"column": 81
} | {
"line": 126,
"column": 81
} | [
{
"pp": "J : Type u_1\ninst✝ : LinearOrder J\nx✝³ : SimplicialThickening J\nx✝² x✝¹ : SimplexCategoryᵒᵖ\nx✝ : x✝² ⟶ x✝¹\n⊢ ((𝟙_ SSet).map x✝ ≫ ↾fun x ↦ (Functor.const (Fin ((Opposite.unop x✝¹).len + 1))).obj (𝟙 x✝³)) =\n (↾fun x ↦ (Functor.const (Fin ((Opposite.unop x✝²).len + 1))).obj (𝟙 x✝³)) ≫ (nerve (... | [] | by simp; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.SimplicialNerve | {
"line": 128,
"column": 16
} | {
"line": 128,
"column": 28
} | {
"line": 128,
"column": 28
} | [
{
"pp": "J : Type u_1\ninst✝ : LinearOrder J\ni j k : SimplicialThickening J\nx✝² x✝¹ : SimplexCategoryᵒᵖ\nx✝ : x✝² ⟶ x✝¹\n⊢ ((nerve (i ⟶ j) ⊗ nerve (j ⟶ k)).map x✝ ≫ ↾fun x ↦ Functor.prod' x.1 x.2 ⋙ i.compFunctor j k) =\n (↾fun x ↦ Functor.prod' x.1 x.2 ⋙ i.compFunctor j k) ≫ (nerve (i ⟶ k)).map x✝",
"p... | [] | by simp; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.RankNat | {
"line": 58,
"column": 2
} | {
"line": 59,
"column": 5
} | {
"line": 61,
"column": 0
} | [
{
"pp": "X : SSet\nA : X.Subcomplex\nP : A.Pairing\ny : ↑P.II\nhy : Acc P.AncestralRel y\n⊢ P.rank' hy = ⨆ x, P.rank' ⋯ + 1",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"iSup",
"SSet.Subcomplex.N",
"Set.Elem",
"id",
"Subtype",
"instOfNatNat",
"A... | [] | change P.rank' (Acc.intro y fun _ => hy.inv) = _
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.RankNat | {
"line": 58,
"column": 2
} | {
"line": 59,
"column": 5
} | {
"line": 61,
"column": 0
} | [
{
"pp": "X : SSet\nA : X.Subcomplex\nP : A.Pairing\ny : ↑P.II\nhy : Acc P.AncestralRel y\n⊢ P.rank' hy = ⨆ x, P.rank' ⋯ + 1",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"iSup",
"SSet.Subcomplex.N",
"Set.Elem",
"id",
"Subtype",
"instOfNatNat",
"A... | [] | change P.rank' (Acc.intro y fun _ => hy.inv) = _
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.RelativeCellComplex | {
"line": 470,
"column": 81
} | {
"line": 472,
"column": 39
} | {
"line": 474,
"column": 0
} | [
{
"pp": "X : SSet\nA : X.Subcomplex\nP : A.Pairing\nι : Type v\ninst✝³ : LinearOrder ι\nf : P.RankFunction ι\ninst✝² : P.IsProper\ninst✝¹ : SuccOrder ι\ninst✝ : NoMaxOrder ι\nj : ι\n⊢ f.t j ≫ homOfLE ⋯ = f.m j ≫ f.b j",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"SSet.Subcomplex.t... | [] | by
ext c : 1
simp [← cancel_mono (Subcomplex.ι _)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Presentable.Limits | {
"line": 165,
"column": 4
} | {
"line": 167,
"column": 51
} | {
"line": 167,
"column": 51
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\nK : Type u'\ninst✝³ : Category.{v', u'} K\nF : K ⥤ C ⥤ Type w'\nc : Cone F\nhc : IsLimit c\nκ : Cardinal.{w}\ninst✝² : Fact κ.IsRegular\ninst✝¹ : HasLimitsOfShape K (Type w')\nhK : HasCardinalLT (Arrow K) κ\ninst✝ : ∀ (k : K), (F.obj k).IsCardinalAccessible κ\nJ ... | [] | exact ⟨Accessible.Limits.isColimitMapCocone c
(fun Y ↦ isLimitOfPreserves ((evaluation C (Type w')).obj Y) hc) κ hK cX
(fun k ↦ isColimitOfPreserves (F.obj k) hcX)⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Functor.KanExtension.DenseAt | {
"line": 122,
"column": 2
} | {
"line": 122,
"column": 49
} | {
"line": 123,
"column": 2
} | [
{
"pp": "C : Type u₁\nD : Type u₂\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nY : D\nhY : F.DenseAt Y\n⊢ F.isDenseAt.IsClosedUnderIsomorphisms",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Functor",
"CategoryTheory.Fu... | [
"C : Type u₁\nD : Type u₂\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nY : D\nhY : F.DenseAt Y\n⊢ (LeftExtension.mk (𝟭 D) F.rightUnitor.inv).isPointwiseLeftKanExtensionAt.IsClosedUnderIsomorphisms"
] | rw [isDenseAt_eq_isPointwiseLeftKanExtensionAt] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.RelativeCellComplex | {
"line": 497,
"column": 6
} | {
"line": 498,
"column": 10
} | {
"line": 498,
"column": 10
} | [
{
"pp": "case refine_2\nX : SSet\nA : X.Subcomplex\nP : A.Pairing\nι : Type v\ninst✝³ : LinearOrder ι\nf : P.RankFunction ι\ninst✝² : P.IsProper\ninst✝¹ : SuccOrder ι\ninst✝ : NoMaxOrder ι\nj : ι\nx✝ : SimplexCategoryᵒᵖ\nd : ℕ\ny : ↑((f.filtration j).obj (op ⦋d⦌))\nx : f.Cell j\nb : Δ[x.dim + 1] _⦋d⦌\nh : ⟨↑y, ... | [] | rw [← NatTrans.comp_app_apply]
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.RelativeCellComplex | {
"line": 497,
"column": 6
} | {
"line": 498,
"column": 10
} | {
"line": 498,
"column": 10
} | [
{
"pp": "case refine_2\nX : SSet\nA : X.Subcomplex\nP : A.Pairing\nι : Type v\ninst✝³ : LinearOrder ι\nf : P.RankFunction ι\ninst✝² : P.IsProper\ninst✝¹ : SuccOrder ι\ninst✝ : NoMaxOrder ι\nj : ι\nx✝ : SimplexCategoryᵒᵖ\nd : ℕ\ny : ↑((f.filtration j).obj (op ⦋d⦌))\nx : f.Cell j\nb : Δ[x.dim + 1] _⦋d⦌\nh : ⟨↑y, ... | [] | rw [← NatTrans.comp_app_apply]
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Adjunction.ParametrizedLimits | {
"line": 41,
"column": 4
} | {
"line": 46,
"column": 93
} | {
"line": 47,
"column": 4
} | [
{
"pp": "C₁ : Type u_1\nC₂ : Type u_2\nC₃ : Type u_3\ninst✝⁴ : Category.{v_1, u_1} C₁\ninst✝³ : Category.{v_2, u_2} C₂\ninst✝² : Category.{v_3, u_3} C₃\nF : C₁ ⥤ C₂ ⥤ C₃\nG : C₁ᵒᵖ ⥤ C₃ ⥤ C₂\nadj₂ : F ⊣₂ G\nJ : Type u_4\ninst✝¹ : Category.{v_4, u_4} J\nP : J ⥤ C₁ᵒᵖ\ninst✝ : ∀ (X₂ : C₂), PreservesColimit P.leftOp... | [
"C₁ : Type u_1\nC₂ : Type u_2\nC₃ : Type u_3\ninst✝⁴ : Category.{v_1, u_1} C₁\ninst✝³ : Category.{v_2, u_2} C₂\ninst✝² : Category.{v_3, u_3} C₃\nF : C₁ ⥤ C₂ ⥤ C₃\nG : C₁ᵒᵖ ⥤ C₃ ⥤ C₂\nadj₂ : F ⊣₂ G\nJ : Type u_4\ninst✝¹ : Category.{v_4, u_4} J\nP : J ⥤ C₁ᵒᵖ\ninst✝ : ∀ (X₂ : C₂), PreservesColimit P.leftOp (F.flip.obj... | let cocone (s : Cone (P ⋙ G.flip.obj X₃)) :
Cocone (P.leftOp ⋙ F.flip.obj s.pt) :=
{ pt := X₃
ι.app j := adj₂.homEquiv.symm (s.π.app j.unop)
ι.naturality _ _ f := by
simp [← s.w f.unop, dsimp% adj₂.homEquiv_symm_naturality_one (P.map f.unop).unop] } | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.AlgebraicTopology.SimplicialSet.Homology.Basic | {
"line": 44,
"column": 47
} | {
"line": 46,
"column": 16
} | {
"line": 48,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasCoproducts C\ninst✝ : Preadditive C\n⊢ (chainComplexFunctor C).Additive",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"CategoryTheory.Functor",
"ChainComplex",
"HomologicalComplex.instCategory",
"Opposit... | [] | by
dsimp [chainComplexFunctor, SimplicialObject.whiskering]
infer_instance | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.SimplicialSet.Homology.HomologyZero | {
"line": 52,
"column": 4
} | {
"line": 52,
"column": 12
} | {
"line": 53,
"column": 4
} | [
{
"pp": "case neg\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasCoproducts C\ninst✝ : Preadditive C\nX : SSet\nR : C\nn : ℕ\nhn : n = 1\n⊢ (X.chainComplex R).d n 0 ≫ fromChainComplexXZero X R = 0",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits.hasColimit... | [
"case neg\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasCoproducts C\ninst✝ : Preadditive C\nX : SSet\nR : C\n⊢ (X.chainComplex R).d 1 0 ≫ fromChainComplexXZero X R = 0"
] | subst hn | Lean.Elab.Tactic.evalSubst | Lean.Parser.Tactic.subst |
Mathlib.AlgebraicTopology.SimplicialSet.KanComplex.MulStruct | {
"line": 44,
"column": 2
} | {
"line": 49,
"column": 83
} | {
"line": 51,
"column": 0
} | [
{
"pp": "X : SSet\nn : ℕ\nx : X _⦋0⦌\ns : X.PtSimplex n x\nY : SSet\nφ : Y ⟶ Δ[n]\ninst✝ : Y.HasDimensionLT n\n⊢ φ ≫ s.map = const x",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"SSet.Subcomplex.toSSet",
"SSet.instHasDimensionLTToSSetRange",
"Eq.mpr",
"SSet.Subco... | [] | refine (Subcomplex.lift φ ?_) ≫= s.comm
rw [stdSimplex.le_boundary_iff]
intro h
have : IsIso (Subcomplex.range φ).ι := by rw [h]; infer_instance
exact stdSimplex.not_hasDimensionLT n
((hasDimensionLT_iff_of_iso (asIso (Subcomplex.range φ).ι) n).mp inferInstance) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplicialSet.KanComplex.MulStruct | {
"line": 44,
"column": 2
} | {
"line": 49,
"column": 83
} | {
"line": 51,
"column": 0
} | [
{
"pp": "X : SSet\nn : ℕ\nx : X _⦋0⦌\ns : X.PtSimplex n x\nY : SSet\nφ : Y ⟶ Δ[n]\ninst✝ : Y.HasDimensionLT n\n⊢ φ ≫ s.map = const x",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"SSet.Subcomplex.toSSet",
"SSet.instHasDimensionLTToSSetRange",
"Eq.mpr",
"SSet.Subco... | [] | refine (Subcomplex.lift φ ?_) ≫= s.comm
rw [stdSimplex.le_boundary_iff]
intro h
have : IsIso (Subcomplex.range φ).ι := by rw [h]; infer_instance
exact stdSimplex.not_hasDimensionLT n
((hasDimensionLT_iff_of_iso (asIso (Subcomplex.range φ).ι) n).mp inferInstance) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SimplicialSet.NerveAdjunction | {
"line": 73,
"column": 31
} | {
"line": 73,
"column": 50
} | {
"line": 73,
"column": 51
} | [
{
"pp": "n : ℕ\nX Y : Truncated 2\nf₀ : X.obj (op { obj := ⦋0⦌, property := _proof_11 }) → Y.obj (op { obj := ⦋0⦌, property := _proof_11 })\nf₁ : X.obj (op { obj := ⦋1⦌, property := _proof_12 }) → Y.obj (op { obj := ⦋1⦌, property := _proof_12 })\nhδ₁ :\n ∀ (x : X.obj (op { obj := ⦋1⦌, property := _proof_12 }))... | [
"n : ℕ\nX Y : Truncated 2\nf₀ : X.obj (op { obj := ⦋0⦌, property := _proof_11 }) → Y.obj (op { obj := ⦋0⦌, property := _proof_11 })\nf₁ : X.obj (op { obj := ⦋1⦌, property := _proof_12 }) → Y.obj (op { obj := ⦋1⦌, property := _proof_12 })\nhδ₁ :\n ∀ (x : X.obj (op { obj := ⦋1⦌, property := _proof_12 })),\n f₀ ((... | ← δ₂_zero_eq_const, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.IteratedDeriv.ConvergenceOnBall | {
"line": 38,
"column": 2
} | {
"line": 38,
"column": 95
} | {
"line": 39,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nf : 𝕜 → 𝕜\nx : 𝕜\nr : ENNReal\nhr_pos : 0 < r\nh : AnalyticOnNhd 𝕜 f (Metric.eball x r)\np : FormalMultilinearSeries 𝕜 𝕜 𝕜 := FormalMultilinearSeries.ofScalars 𝕜 fun n ↦ iteratedDeriv n f x / ↑n.factorial\nhr : r ≤ p.radius\ng : 𝕜 → 𝕜 := fun t ↦ p.sum (t - x)... | [
"𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nf : 𝕜 → 𝕜\nx : 𝕜\nr : ENNReal\nhr_pos : 0 < r\nh : AnalyticOnNhd 𝕜 f (Metric.eball x r)\np : FormalMultilinearSeries 𝕜 𝕜 𝕜 := FormalMultilinearSeries.ofScalars 𝕜 fun n ↦ iteratedDeriv n f x / ↑n.factorial\nhr : r ≤ p.radius\ng : 𝕜 → 𝕜 := fun t ↦ p.sum (t - x)\nhg : HasFP... | replace hg' : AnalyticOnNhd 𝕜 g (Metric.eball x r) := hg'.mono (Metric.eball_subset_eball hr) | Lean.Elab.Tactic.evalReplace | Lean.Parser.Tactic.replace |
Mathlib.Analysis.AbsoluteValue.Equivalence | {
"line": 273,
"column": 6
} | {
"line": 273,
"column": 37
} | {
"line": 274,
"column": 6
} | [
{
"pp": "case refine_2.inr.inr.inl\nR : Type u_1\nS : Type u_2\ninst✝⁷ : Field R\ninst✝⁶ : Field S\ninst✝⁵ : LinearOrder S\ninst✝⁴ : TopologicalSpace S\ninst✝³ : IsStrictOrderedRing S\ninst✝² : Archimedean S\ninst✝¹ : OrderTopology S\nι✝ : Type u_3\ninst✝ : Finite ι✝\nv✝ : ι✝ → AbsoluteValue R S\nthis : Fintype... | [
"case refine_2.inr.inr.inl\nR : Type u_1\nS : Type u_2\ninst✝⁷ : Field R\ninst✝⁶ : Field S\ninst✝⁵ : LinearOrder S\ninst✝⁴ : TopologicalSpace S\ninst✝³ : IsStrictOrderedRing S\ninst✝² : Archimedean S\ninst✝¹ : OrderTopology S\nι✝ : Type u_3\ninst✝ : Finite ι✝\nv✝ : ι✝ → AbsoluteValue R S\nthis : Fintype ι✝\nP : (ι ... | refine ⟨a, ha.1, fun k hk ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.AbsoluteValue.Equivalence | {
"line": 313,
"column": 2
} | {
"line": 313,
"column": 31
} | {
"line": 314,
"column": 2
} | [
{
"pp": "F : Type u_1\ninst✝ : Field F\nv w : AbsoluteValue F ℝ\nh : v.IsEquiv w\na : F\nha₀ : a ≠ 0\nha₁ : v a ≠ 1\nb : F\nhb₀ : b ≠ 0\nhb₁ : v b ≠ 1\nh_ne : log (v b) / log (w b) ≠ log (v a) / log (w a)\nha : 1 < v a\nhb : 1 < v b\nh_lt : log (v b) / log (w b) < log (v a) / log (w a)\n⊢ False",
"ppTerm": ... | [
"F : Type u_1\ninst✝ : Field F\nv w : AbsoluteValue F ℝ\nh : v.IsEquiv w\na : F\nha₀ : a ≠ 0\nha₁ : v a ≠ 1\nb : F\nhb₀ : b ≠ 0\nhb₁ : v b ≠ 1\nh_ne : log (v b) / log (w b) ≠ log (v a) / log (w a)\nha : 1 < v a\nhb : 1 < v b\nh_lt : log (v b) / log (w b) < log (v a) / log (w a)\nhwa : 1 < w a\n⊢ False"
] | have hwa := h.one_lt_iff.1 ha | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.AlgebraicTopology.SimplicialSet.NerveAdjunction | {
"line": 167,
"column": 4
} | {
"line": 169,
"column": 43
} | {
"line": 170,
"column": 4
} | [
{
"pp": "case refine_2.succ.zero\nX Y : Truncated 2\nf₀ : X.obj (op { obj := ⦋0⦌, property := _proof_11 }) → Y.obj (op { obj := ⦋0⦌, property := _proof_11 })\nf₁ : X.obj (op { obj := ⦋1⦌, property := _proof_12 }) → Y.obj (op { obj := ⦋1⦌, property := _proof_12 })\nhδ₁ :\n ∀ (x : X.obj (op { obj := ⦋1⦌, propert... | [
"case refine_2.succ.succ\nX Y : Truncated 2\nf₀ : X.obj (op { obj := ⦋0⦌, property := _proof_11 }) → Y.obj (op { obj := ⦋0⦌, property := _proof_11 })\nf₁ : X.obj (op { obj := ⦋1⦌, property := _proof_12 }) → Y.obj (op { obj := ⦋1⦌, property := _proof_12 })\nhδ₁ :\n ∀ (x : X.obj (op { obj := ⦋1⦌, property := _proof_... | · fin_cases i
· ext; apply hσ'₀ f₀ f₁ hδ₁ hδ₀ hσ hY
· ext; apply hσ'₁ f₀ f₁ hδ₁ hδ₀ hσ hY | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Meromorphic.Basic | {
"line": 78,
"column": 2
} | {
"line": 78,
"column": 24
} | {
"line": 79,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx : 𝕜\nf g : 𝕜 → E\nhg : MeromorphicAt g x\nm : ℕ\nhf : AnalyticAt 𝕜 (fun z ↦ (z - x) ^ m • f z) x\n⊢ MeromorphicAt (f + g) x",
"ppTerm": "?m.38",
"assigned": true,
... | [
"𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx : 𝕜\nf g : 𝕜 → E\nm : ℕ\nhf : AnalyticAt 𝕜 (fun z ↦ (z - x) ^ m • f z) x\nn : ℕ\nhg : AnalyticAt 𝕜 (fun z ↦ (z - x) ^ n • g z) x\n⊢ MeromorphicAt (f + g) x"
] | rcases hg with ⟨n, hg⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.Meromorphic.Basic | {
"line": 93,
"column": 2
} | {
"line": 93,
"column": 24
} | {
"line": 94,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\nR : Type u_4\ninst✝⁴ : NormedRing R\ninst✝³ : Module R E\ninst✝² : IsBoundedSMul R E\nx : 𝕜\ninst✝¹ : NormedAlgebra 𝕜 R\ninst✝ : IsScalarTower 𝕜 R E\nf : 𝕜 → R\ng : 𝕜 → E\nhg... | [
"𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\nR : Type u_4\ninst✝⁴ : NormedRing R\ninst✝³ : Module R E\ninst✝² : IsBoundedSMul R E\nx : 𝕜\ninst✝¹ : NormedAlgebra 𝕜 R\ninst✝ : IsScalarTower 𝕜 R E\nf : 𝕜 → R\ng : 𝕜 → E\nm : ℕ\nhf : An... | rcases hg with ⟨n, hg⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.Analytic.Order | {
"line": 358,
"column": 97
} | {
"line": 370,
"column": 32
} | {
"line": 372,
"column": 0
} | [
{
"pp": "𝕜 : Type u_3\nE : Type u_4\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : CompleteSpace E\nf : 𝕜 → E\nz₀ : 𝕜\nhf : AnalyticAt 𝕜 f z₀\nk n : ℕ\ninst✝ : CharZero 𝕜\n⊢ ↑n = analyticOrderAt f z₀ → n ≠ 0 → k ≤ n → analyticOrderAt (deriv^[k] f) z... | [] | by
induction k generalizing n with
| zero => exact fun Hn Hpos Hk ↦ Hn.symm
| succ n' hk =>
intro Hn Hpos Hk
rw [Function.iterate_succ']
have horder : analyticOrderAt (deriv^[n'] f) z₀ = (n - n'.succ) + 1 := by
refine (hk Hn Hpos (by lia)).trans ?_
have : (n - n'.succ) + 1 = n - n' := by g... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.OrdinaryHypergeometric | {
"line": 163,
"column": 2
} | {
"line": 163,
"column": 36
} | {
"line": 164,
"column": 2
} | [
{
"pp": "𝕂 : Type u_1\n𝔸 : Type u_2\ninst✝² : RCLike 𝕂\ninst✝¹ : NormedDivisionRing 𝔸\ninst✝ : NormedAlgebra 𝕂 𝔸\na b c : 𝕂\nn : ℕ\n⊢ ordinaryHypergeometricSeries 𝔸 a b c n = 0 ↔ ∃ k < n, ↑k = -a ∨ ↑k = -b ∨ ↑k = -c",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"NegZeroClas... | [
"case refine_1\n𝕂 : Type u_1\n𝔸 : Type u_2\ninst✝² : RCLike 𝕂\ninst✝¹ : NormedDivisionRing 𝔸\ninst✝ : NormedAlgebra 𝕂 𝔸\na b c : 𝕂\nn : ℕ\nh : ordinaryHypergeometricSeries 𝔸 a b c n = 0\n⊢ ∃ k < n, ↑k = -a ∨ ↑k = -b ∨ ↑k = -c",
"case refine_2\n𝕂 : Type u_1\n𝔸 : Type u_2\ninst✝² : RCLike 𝕂\ninst✝¹ : Nor... | refine ⟨fun h ↦ ?_, fun zero ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Analytic.Order | {
"line": 389,
"column": 2
} | {
"line": 394,
"column": 76
} | {
"line": 396,
"column": 0
} | [
{
"pp": "case convert_2\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : CharZero 𝕜\ninst✝ : CompleteSpace E\nf : 𝕜 → E\nhf : AnalyticAt 𝕜 f 0\nn : ℕ\nthis : AnalyticAt 𝕜 (fun z ↦ ∑ i ∈ Finset.range n, (z ^ i / ↑i.factorial... | [] | · rw [natCast_le_analyticOrderAt_iff_iteratedDeriv_eq_zero (hf.fun_sub this)]
intro i hi
rw [iteratedDeriv_fun_sub (AnalyticAt.contDiffAt <| by fun_prop) this.contDiffAt]
simp (disch := fun_prop) only [iteratedDeriv_fun_sum, iteratedDeriv_smul_const,
iteratedDeriv_div_const, iteratedDeriv_fun_pow_zero... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Meromorphic.Basic | {
"line": 367,
"column": 4
} | {
"line": 367,
"column": 86
} | {
"line": 368,
"column": 4
} | [
{
"pp": "case refine_2\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx : 𝕜\nf : 𝕜 → E\n⊢ (∃ n g, AnalyticAt 𝕜 g x ∧ ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z = (z - x) ^ n • g z) → MeromorphicAt f x",
"ppTerm": "?refine_2",
"assigned": t... | [
"case refine_2\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx : 𝕜\nf : 𝕜 → E\nx✝ : ∃ n g, AnalyticAt 𝕜 g x ∧ ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z = (z - x) ^ n • g z\nn : ℤ\ng : 𝕜 → E\nhg_an : AnalyticAt 𝕜 g x\nhg_eq : ∀ᶠ (z : 𝕜) in 𝓝[≠] x... | refine fun ⟨n, g, hg_an, hg_eq⟩ ↦ MeromorphicAt.congr ?_ (EventuallyEq.symm hg_eq) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Analytic.Order | {
"line": 468,
"column": 37
} | {
"line": 468,
"column": 66
} | {
"line": 468,
"column": 67
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : CharZero 𝕜\ninst✝ : CompleteSpace E\nz₀ : 𝕜\nn : ℕ\nf : 𝕜 → E\nhf : AnalyticAt 𝕜 f z₀\nIH : analyticOrderAt (deriv f) z₀ = ↑n ↔ (∀ k < n, iteratedDeriv (k + 1) f z₀ =... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : CharZero 𝕜\ninst✝ : CompleteSpace E\nz₀ : 𝕜\nn : ℕ\nf : 𝕜 → E\nhf : AnalyticAt 𝕜 f z₀\nIH : analyticOrderAt (deriv f) z₀ = ↑n ↔ (∀ k < n, iteratedDeriv (k + 1) f z₀ = 0) ∧ iterat... | ← hf.analyticOrderAt_ne_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Binomial | {
"line": 122,
"column": 69
} | {
"line": 122,
"column": 78
} | {
"line": 122,
"column": 79
} | [
{
"pp": "R : Type u_1\ninst✝² : AddCommMonoid R\ninst✝¹ : Pow R ℕ\ninst✝ : BinomialRing R\nr : R\n⊢ X.smeval r = Nat.factorial 1 • r ^ 1",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"congrArg",
"AddMonoid.toNSMul",
"Module.toMulAction... | [
"R : Type u_1\ninst✝² : AddCommMonoid R\ninst✝¹ : Pow R ℕ\ninst✝ : BinomialRing R\nr : R\n⊢ r ^ 1 = Nat.factorial 1 • r ^ 1"
] | smeval_X, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Binomial | {
"line": 222,
"column": 85
} | {
"line": 222,
"column": 98
} | {
"line": 223,
"column": 6
} | [
{
"pp": "case succ\nR : Type u_1\ninst✝² : NonAssocRing R\ninst✝¹ : Pow R ℕ\ninst✝ : NatPowAssoc R\nn k : ℕ\nih : (descPochhammer ℤ k).smeval ↑n = ↑(n.descFactorial k)\n⊢ ↑(n.descFactorial k) * (X - ↑k).smeval ↑n = ↑(n.descFactorial k * (n - k))",
"ppTerm": "?succ",
"assigned": true,
"usedConstants"... | [
"case succ\nR : Type u_1\ninst✝² : NonAssocRing R\ninst✝¹ : Pow R ℕ\ninst✝ : NatPowAssoc R\nn k : ℕ\nih : (descPochhammer ℤ k).smeval ↑n = ↑(n.descFactorial k)\n⊢ ↑(n.descFactorial k) * (X - ↑k).smeval ↑n = ↑(n.descFactorial k) * ↑(n - k)"
] | Nat.cast_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Binomial | {
"line": 223,
"column": 18
} | {
"line": 223,
"column": 27
} | {
"line": 223,
"column": 28
} | [
{
"pp": "case succ\nR : Type u_1\ninst✝² : NonAssocRing R\ninst✝¹ : Pow R ℕ\ninst✝ : NatPowAssoc R\nn k : ℕ\nih : (descPochhammer ℤ k).smeval ↑n = ↑(n.descFactorial k)\n⊢ ↑(n.descFactorial k) * (X.smeval ↑n - (↑k).smeval ↑n) = ↑(n.descFactorial k) * ↑(n - k)",
"ppTerm": "?succ",
"assigned": true,
"u... | [
"case succ\nR : Type u_1\ninst✝² : NonAssocRing R\ninst✝¹ : Pow R ℕ\ninst✝ : NatPowAssoc R\nn k : ℕ\nih : (descPochhammer ℤ k).smeval ↑n = ↑(n.descFactorial k)\n⊢ ↑(n.descFactorial k) * (↑n ^ 1 - (↑k).smeval ↑n) = ↑(n.descFactorial k) * ↑(n - k)"
] | smeval_X, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Binomial | {
"line": 300,
"column": 72
} | {
"line": 300,
"column": 81
} | {
"line": 300,
"column": 82
} | [
{
"pp": "n : ℕ\n⊢ -↑(n + 1) * (ascPochhammer ℕ n).smeval (X.smeval (-↑(n + 1)) + smeval 1 (-↑(n + 1))) =\n (-1) ^ (n + 1) * ↑(n + 1).factorial",
"ppTerm": "?m.125",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Int.instAddCommMonoid",
"Polynomial.instOne",
"HMul.hMul",
... | [
"n : ℕ\n⊢ -↑(n + 1) * (ascPochhammer ℕ n).smeval ((-↑(n + 1)) ^ 1 + smeval 1 (-↑(n + 1))) = (-1) ^ (n + 1) * ↑(n + 1).factorial"
] | smeval_X, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Binomial | {
"line": 304,
"column": 21
} | {
"line": 304,
"column": 34
} | {
"line": 304,
"column": 35
} | [
{
"pp": "n : ℕ\n⊢ (-1) ^ n * -1 * (↑n + 1) * ↑n.factorial = (-1) ^ n * -1 * ↑(n.succ * n.factorial)",
"ppTerm": "?m.241",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Semigroup.toMul",
"HMul.hMul",
"Monoid.toMulOneClass",
... | [
"n : ℕ\n⊢ (-1) ^ n * -1 * (↑n + 1) * ↑n.factorial = (-1) ^ n * -1 * (↑n.succ * ↑n.factorial)"
] | Nat.cast_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Binomial | {
"line": 309,
"column": 56
} | {
"line": 309,
"column": 65
} | {
"line": 309,
"column": 66
} | [
{
"pp": "n : ℕ\n⊢ (ascPochhammer ℕ n).smeval (-↑n) * (X.smeval (-↑n) + (↑n).smeval (-↑n)) = 0",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Int.instAddCommMonoid",
"HMul.hMul",
"congrArg",
"ascPochhammer",
"Module.toMulActionWithZero",
... | [
"n : ℕ\n⊢ (ascPochhammer ℕ n).smeval (-↑n) * ((-↑n) ^ 1 + (↑n).smeval (-↑n)) = 0"
] | smeval_X, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Binomial | {
"line": 315,
"column": 18
} | {
"line": 315,
"column": 47
} | {
"line": 315,
"column": 47
} | [
{
"pp": "n : ℕ\n⊢ (ascPochhammer ℕ (n + 1)).smeval (-↑n) = 0",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Int.instAddCommMonoid",
"congrArg",
"ascPochhammer",
"Module.toMulActionWithZero",
"id",
"Int.instNegInt",
"instOfNatNat",... | [
"n : ℕ\n⊢ 0 = 0"
] | smeval_ascPochhammer_succ_neg | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Binomial | {
"line": 393,
"column": 43
} | {
"line": 393,
"column": 52
} | {
"line": 393,
"column": 53
} | [
{
"pp": "R : Type u_1\ninst✝³ : NonAssocRing R\ninst✝² : Pow R ℕ\ninst✝¹ : BinomialRing R\ninst✝ : NatPowAssoc R\nr : R\nn : ℕ\n⊢ (ascPochhammer ℤ n).smeval (X.smeval r + (1 - ↑n).smeval r) = (ascPochhammer ℕ n).smeval (r - ↑n + 1)",
"ppTerm": "?m.66",
"assigned": true,
"usedConstants": [
"Eq.... | [
"R : Type u_1\ninst✝³ : NonAssocRing R\ninst✝² : Pow R ℕ\ninst✝¹ : BinomialRing R\ninst✝ : NatPowAssoc R\nr : R\nn : ℕ\n⊢ (ascPochhammer ℤ n).smeval (r ^ 1 + (1 - ↑n).smeval r) = (ascPochhammer ℕ n).smeval (r - ↑n + 1)"
] | smeval_X, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Binomial | {
"line": 472,
"column": 75
} | {
"line": 472,
"column": 84
} | {
"line": 473,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝³ : NonAssocRing R\ninst✝² : Pow R ℕ\ninst✝¹ : BinomialRing R\ninst✝ : NatPowAssoc R\nr : R\nn k : ℕ\nhkn : k ≤ n\n⊢ n.choose k • ((descPochhammer ℤ k).smeval r * (descPochhammer ℤ (n - k)).smeval (X.smeval r - (↑k).smeval r)) =\n n.choose k • ((descPochhammer ℤ k).smeval r * (des... | [
"R : Type u_1\ninst✝³ : NonAssocRing R\ninst✝² : Pow R ℕ\ninst✝¹ : BinomialRing R\ninst✝ : NatPowAssoc R\nr : R\nn k : ℕ\nhkn : k ≤ n\n⊢ n.choose k • ((descPochhammer ℤ k).smeval r * (descPochhammer ℤ (n - k)).smeval (r ^ 1 - (↑k).smeval r)) =\n n.choose k • ((descPochhammer ℤ k).smeval r * (descPochhammer ℤ (n ... | smeval_X, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Binomial | {
"line": 498,
"column": 37
} | {
"line": 498,
"column": 46
} | {
"line": 498,
"column": 47
} | [
{
"pp": "case succ\nR : Type u_1\ninst✝ : Ring R\nr s : R\nh : Commute r s\nk : ℕ\nih :\n (descPochhammer ℤ k).smeval (r + s) =\n ∑ ij ∈ antidiagonal k, ↑(k.choose ij.1) * ((descPochhammer ℤ ij.1).smeval r * (descPochhammer ℤ ij.2).smeval s)\n⊢ (X.smeval (r + s) - (↑k).smeval (r + s)) * (descPochhammer ℤ k)... | [
"case succ\nR : Type u_1\ninst✝ : Ring R\nr s : R\nh : Commute r s\nk : ℕ\nih :\n (descPochhammer ℤ k).smeval (r + s) =\n ∑ ij ∈ antidiagonal k, ↑(k.choose ij.1) * ((descPochhammer ℤ ij.1).smeval r * (descPochhammer ℤ ij.2).smeval s)\n⊢ ((r + s) ^ 1 - (↑k).smeval (r + s)) * (descPochhammer ℤ k).smeval (r + s) =... | smeval_X, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Binomial | {
"line": 528,
"column": 4
} | {
"line": 528,
"column": 17
} | {
"line": 528,
"column": 18
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Ring R\ninst✝ : BinomialRing R\nr s : R\nk : ℕ\nh : Commute r s\nx : ℕ × ℕ\nhx : x ∈ antidiagonal k\n⊢ ↑(k.choose x.1) * ((descPochhammer ℤ x.1).smeval r * (descPochhammer ℤ x.2).smeval s) =\n ↑(k.choose x.1 * (x.1.factorial * x.2.factorial)) * (choose r x.1 * choose s x.2)",
... | [
"R : Type u_1\ninst✝¹ : Ring R\ninst✝ : BinomialRing R\nr s : R\nk : ℕ\nh : Commute r s\nx : ℕ × ℕ\nhx : x ∈ antidiagonal k\n⊢ ↑(k.choose x.1) * ((descPochhammer ℤ x.1).smeval r * (descPochhammer ℤ x.2).smeval s) =\n ↑(k.choose x.1) * ↑(x.1.factorial * x.2.factorial) * (choose r x.1 * choose s x.2)"
] | Nat.cast_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Binomial | {
"line": 528,
"column": 18
} | {
"line": 528,
"column": 31
} | {
"line": 528,
"column": 32
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Ring R\ninst✝ : BinomialRing R\nr s : R\nk : ℕ\nh : Commute r s\nx : ℕ × ℕ\nhx : x ∈ antidiagonal k\n⊢ ↑(k.choose x.1) * ((descPochhammer ℤ x.1).smeval r * (descPochhammer ℤ x.2).smeval s) =\n ↑(k.choose x.1) * ↑(x.1.factorial * x.2.factorial) * (choose r x.1 * choose s x.2)",... | [
"R : Type u_1\ninst✝¹ : Ring R\ninst✝ : BinomialRing R\nr s : R\nk : ℕ\nh : Commute r s\nx : ℕ × ℕ\nhx : x ∈ antidiagonal k\n⊢ ↑(k.choose x.1) * ((descPochhammer ℤ x.1).smeval r * (descPochhammer ℤ x.2).smeval s) =\n ↑(k.choose x.1) * (↑x.1.factorial * ↑x.2.factorial) * (choose r x.1 * choose s x.2)"
] | Nat.cast_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Pow.Deriv | {
"line": 354,
"column": 27
} | {
"line": 354,
"column": 48
} | {
"line": 354,
"column": 49
} | [
{
"pp": "case e'_12\np : ℝ × ℝ\nhp : 0 < p.1\nthis : (fun x ↦ x.1 ^ x.2) =ᶠ[𝓝 p] fun x ↦ rexp (log x.1 * x.2)\ne_4✝ : Prod.instAddCommGroup = Prod.normedAddCommGroup.toAddCommGroup\ne_5✝ : Prod.instModule ≍ Prod.normedSpace.toModule\ne_6✝ : instTopologicalSpaceProd = PseudoMetricSpace.toUniformSpace.toTopologi... | [
"case e'_12\np : ℝ × ℝ\nhp : 0 < p.1\nthis : (fun x ↦ x.1 ^ x.2) =ᶠ[𝓝 p] fun x ↦ rexp (log x.1 * x.2)\ne_4✝ : Prod.instAddCommGroup = Prod.normedAddCommGroup.toAddCommGroup\ne_5✝ : Prod.instModule ≍ Prod.normedSpace.toModule\ne_6✝ : instTopologicalSpaceProd = PseudoMetricSpace.toUniformSpace.toTopologicalSpace\n⊢ ... | ← rpow_def_of_pos hp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Pow.Deriv | {
"line": 422,
"column": 4
} | {
"line": 422,
"column": 53
} | {
"line": 423,
"column": 4
} | [
{
"pp": "r : ℝ\nhr : r < 1\nhr' : r ≠ 0\nh : DifferentiableAt ℝ (fun x ↦ x ^ r) 0\ny : ℝ := deriv (fun x ↦ x ^ r) 0\nx : ℝ\nhx : 0 < x\n⊢ x ^ r = x * x ^ (r - 1)",
"ppTerm": "?m.121",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"Real.instPow",
... | [
"r : ℝ\nhr : r < 1\nhr' : r ≠ 0\nh : DifferentiableAt ℝ (fun x ↦ x ^ r) 0\ny : ℝ := deriv (fun x ↦ x ^ r) 0\nx : ℝ\nhx : 0 < x\n⊢ x ^ 1 * x ^ (r - 1) = x * x ^ (r - 1)"
] | nth_rw 1 [← add_sub_cancel 1 r, Real.rpow_add hx] | Mathlib.Tactic._aux_Mathlib_Tactic_NthRewrite___macroRules_Mathlib_Tactic_tacticNth_rw______1 | Mathlib.Tactic.tacticNth_rw_____ |
Mathlib.Analysis.Analytic.Polynomial | {
"line": 66,
"column": 2
} | {
"line": 66,
"column": 38
} | {
"line": 67,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : CommSemiring A\nz : E\ninst✝² : NormedCommRing B\ninst✝¹ : NormedAlgebra 𝕜 B\ninst✝ : Algebra A B\nσ : Type u_5\nf : E → σ → B\nhf : ∀ (i : σ... | [
"𝕜 : Type u_1\nE : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : CommSemiring A\nz : E\ninst✝² : NormedCommRing B\ninst✝¹ : NormedAlgebra 𝕜 B\ninst✝ : Algebra A B\nσ : Type u_5\nf : E → σ → B\nhf : ∀ (i : σ), AnalyticA... | · simp_rw [map_add]; exact hp.add hq | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
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